4(x^2+21x-197.5)=0

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Solution for 4(x^2+21x-197.5)=0 equation: 


Simplifying
4(x2 + 21x + -197.5) = 0

Reorder the terms:
4(-197.5 + 21x + x2) = 0
(-197.5 * 4 + 21x * 4 + x2 * 4) = 0
(-790 + 84x + 4x2) = 0

Solving
-790 + 84x + 4x2 = 0

Solving for variable 'x'.

Factor out the Greatest Common Factor (GCF), '2'.
2(-395 + 42x + 2x2) = 0

Ignore the factor 2.

Subproblem 1
Set the factor '(-395 + 42x + 2x2)' equal to zero and attempt to solve:

Simplifying
-395 + 42x + 2x2 = 0

Solving
-395 + 42x + 2x2 = 0

Begin completing the square.  Divide all terms by
2 the coefficient of the squared term: 

Divide each side by '2'.
-197.5 + 21x + x2 = 0

Move the constant term to the right:

Add '197.5' to each side of the equation.
-197.5 + 21x + 197.5 + x2 = 0 + 197.5

Reorder the terms:
-197.5 + 197.5 + 21x + x2 = 0 + 197.5

Combine like terms: -197.5 + 197.5 = 0.0
0.0 + 21x + x2 = 0 + 197.5
21x + x2 = 0 + 197.5

Combine like terms: 0 + 197.5 = 197.5
21x + x2 = 197.5

The x term is 21x.  Take half its coefficient (10.5).
Square it (110.25) and add it to both sides.

Add '110.25' to each side of the equation.
21x + 110.25 + x2 = 197.5 + 110.25

Reorder the terms:
110.25 + 21x + x2 = 197.5 + 110.25

Combine like terms: 197.5 + 110.25 = 307.75
110.25 + 21x + x2 = 307.75

Factor a perfect square on the left side:
(x + 10.5)(x + 10.5) = 307.75

Calculate the square root of the right side: 17.542804793

Break this problem into two subproblems by setting 
(x + 10.5) equal to 17.542804793 and -17.542804793.


Subproblem 1
x + 10.5 = 17.542804793

Simplifying
x + 10.5 = 17.542804793

Reorder the terms:
10.5 + x = 17.542804793

Solving
10.5 + x = 17.542804793

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-10.5' to each side of the equation.
10.5 + -10.5 + x = 17.542804793 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + x = 17.542804793 + -10.5
x = 17.542804793 + -10.5

Combine like terms: 17.542804793 + -10.5 = 7.042804793
x = 7.042804793

Simplifying
x = 7.042804793


Subproblem 2
x + 10.5 = -17.542804793

Simplifying
x + 10.5 = -17.542804793

Reorder the terms:
10.5 + x = -17.542804793

Solving
10.5 + x = -17.542804793

Solving for variable 'x'.

Move all terms containing x to the left, all other terms to the right.

Add '-10.5' to each side of the equation.
10.5 + -10.5 + x = -17.542804793 + -10.5

Combine like terms: 10.5 + -10.5 = 0.0
0.0 + x = -17.542804793 + -10.5
x = -17.542804793 + -10.5

Combine like terms: -17.542804793 + -10.5 = -28.042804793
x = -28.042804793

Simplifying
x = -28.042804793


Solution
The solution to the problem is based on the solutions
from the subproblems.
x = {7.042804793, -28.042804793}
Solution
x = {7.042804793, -28.042804793}

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